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arXiv:1008.2324 · Physical Review A

Creation of topological states of a Bose-Einstein condensate in a square plaquette of four optical traps

Tomasz Świsłocki, Tomasz Sowiński, Mirosław Brewczyk, Mariusz Gajda

Published Aug 16, 2011• 2 citations
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Abstract

We study a square plaquette of four optical microtraps containing ultracold ${}^{87}$Rb atoms in the $F=1$ hyperfine state. In the presence of an external resonant magnetic field, the dipolar interactions couple the initial ${m}_{F}=1$ component to other Zeeman sublevels. This process is a generalization of the Einstein--de Haas effect to the case when the external potential has only ${C}_{4}$ point symmetry. We observe that vortex structures appear in the initially empty ${m}_{F}=0$ state. Topological properties of this state arise due to competition between the local axial symmetry of the individual trap and the discrete symmetry of the plaquette. For deep microtraps vortices are localized at individual sites, whereas for shallow traps only one discrete vortex appears in the plaquette. States created in these two opposite cases have different topological properties related to ${C}_{4}$ point symmetry.

DOI
10.1103/physreva.84.023625

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We study a square plaquette of four optical microtraps containing ultracold 87{}^{87}87Rb atoms in the F=1F=1F=1 hyperfine state. In the presence of an external resonant magnetic field, the dipolar interactions couple the initial mF=1{m}_{F}=1mF​=1 component to other Zeeman sublevels. This process is a generalization of the Einstein--de Haas effect to the case when the external potential has only C4{C}_{4}C4​ point symmetry. We observe that vortex structures appear in the initially empty mF=0{m}_{F}=0mF​=0 state. Topological properties of this state arise due to competition between the local axial symmetry of the individual trap and the discrete symmetry of the plaquette. For deep microtraps vortices are localized at individual sites, whereas for shallow traps only one discrete vortex appears in the plaquette. States created in these two opposite cases have different topological properties related to C4{C}_{4}C4​ point symmetry.
We present a simple mechanism to produce vortices at any desired spatial locations in harmonically trapped Bose-Einstein condensates (BECs) with multicomponent spin states coupled to external transverse and axial magnetic fields. The vortices appear at the spatial points where the spin-transverse field interaction vanishes and, depending on the multipolar magnetic field order, the vortices can acquire different predictable topological charges. We explicitly demonstrate our findings, both numerically and analytically, by analyzing a 2D BEC via the Gross-Pitaevskii equation for atomic systems with either two or three internal states. We further show that by a spontaneous symmetry-breaking mechanism, vortices can appear in any spin component, unless symmetry is externally broken at the outset by an axial field. We suggest that this scenario may be tested using an ultracold gas of 87{}^{87}87Rb occupying all three F=1F=1F=1 states in an optical trap.
Einstein-de Haas effect for a 87^{87}87Rb atoms condensate
We consider a condensate of 87^{87}87Rb atoms in an F=1 hyperfine state confined in an optical dipole trap. Assuming the magnetic moments of all atoms are initially aligned along the magnetic field we observe, after the field's direction is reversed, a transfer of atoms to other Zeeman states due to the dipolar interaction. We find that there exist resonances which amplify the effect of dipolar interaction. When the resonance condition is satisfied the transfer of atoms to mF=0,−1m_F=0,-1mF​=0,−1 states might be of the order of the initial population of mF=1m_F=1mF​=1 component. While changing the magnetic moments during the transfer, the atomsacquire an orbital angular momentum and start to circulate around the center of the trap. This is a realization of the famous Einstein-de Haas effect in systems of cold gases. Due to the dipolar resonances, it becomes possible to observe this phenomenon even in a condensate of 87^{87}87Rb (F=1) atoms that have very small magnetic moment.
We theoretically study a spinor condensate of 87Rb^{87}\mathrm{Rb}87Rb atoms in a F=1F=1F=1 hyperfine state confined in an optical dipole trap. Putting initially all atoms in an mF=1{m}_{F}=1mF​=1, component we observe a significant transfer of atoms to other, initially empty Zeeman states exclusively due to dipolar forces. Because of conservation of a total angular momentum the atoms going to other Zeeman components acquire an orbital angular momentum and circulate around the center of the trap. This is a realization of the Einstein-de Haas effect in a system of cold gases. We show that the transfer of atoms via dipolar interactions is possible only when the energies of the initial and the final sates are equal. This condition can be fulfilled utilizing a resonant external magnetic field, which tunes energies of involved states via the linear Zeeman effect. We found that there are many final states of different spatial density, which can be tuned selectively to the initial state. We show a simple model explaining high selectivity and controllability of weak dipolar interactions in the condensate of 87Rb^{87}\mathrm{Rb}87Rb atoms.
It is shown that a vortex can be continuously created in a Bose-Einstein condensate with hyperfine spin F=2 in a Ioffe-Pritchard trap by reversing the axial magnetic field adiabatically. It may be speculated that the condensate cannot be confined in the trap since the weak-field seeking state makes transitions to the neutral and the strong-field seeking states due to the degeneracy of these states along the vortex axis when the axial field vanishes. We have solved the Gross-Pitaevskii equation numerically with given external magnetic fields to show that this is not the case. It is shown that a considerable fraction of the condensate remains in the trap even when the axial field is reversed rather slowly. This scenario is also analysed in the presence of an optical plug along the vortex axis. Then the condensate remains within the Fz=2F_z=2Fz​=2 manifold, with respect to the local magnetic field, throughout the formation of a vortex and hence the loss of atoms does not take place.
We numerically investigate a condensate of 87Rb atoms in an F=1 hyperfine state confined in an optical dipole trap. Assuming the magnetic moments of all atoms are initially aligned along the magnetic field we observe, after the field's direction is reversed, a transfer of atoms to other Zeeman states. Such transfer is allowed by the dipolar interaction. The atoms in m_F=0, −1 start to circulate around the center of the trap. We find resonances which amplify this phenomenon making it observable even in rubidium condensates.
We studied the dynamics of quadruply quantized vortices in 87 Rb Bose–Einstein condensates. Vortices were created in magnetically trapped condensates with hyperfine spin F = 2 by employing topological phase imprinting technique, wherein the direction of atomic spin is adiabatically reversed by applying a bias magnetic field. Vortices were observed for a holding time of up to 10 ms. Disappearance of the vortices was attributed to considerable expansion and excitation of the condensates, which were caused by the distortion of the magnetic potential. In order to observe the long-term behavior of the vortices, we transferred the condensates to a crossed-type optical dipole force trap after creating the vortices. In this case, the vortices were observed for a holding time of up to 22 ms. We also observed density profiles, which indicated the presence of split vortices.